 Solution

Let $\theta ={\xi }^{\ast }$, and work with the set structure $V_{\theta }\models \mathrm{ZFC}$. Its ordinal height is an infinite limit: a transitive model of ZFC has no largest ordinal, since it can take the successor of each ordinal it contains. Thus $\mathrm{cf}(\theta )\ge \omega$.

Suppose instead that $\mathrm{cf}(\theta )>\omega$. Enumerate all first-order formulas, and close each finite initial collection under subformulas. Applying Lévy reflection theorem inside $V_{\theta }$, choose a strictly increasing sequence ${\beta }_m<\theta$ such that $V_{{\beta }_m}$ agrees with $V_{\theta }$ on the first $m$ collections. Take $\eta ={\sup }_m{\beta }_m<\theta$. The internal rank levels here are the actual rank levels, because $V_{\theta }$ is a transitive rank model.

If $V_{\theta }\models \exists x\psi (x,\vec{a})$ with $\vec{a}\in V_{\eta }$, choose $m$ large enough to include this formula and all the parameters in $V_{{\beta }_m}$. Reflection supplies a witness in $V_{{\beta }_m}\subseteq V_{\eta }$. The Tarski-Vaught test therefore gives $V_{\eta }\prec V_{\theta }$. In particular $V_{\eta }$ satisfies all of ZFC, contrary to the minimality of $\theta$.

ConsequentlyThe countable enumeration is of formulas, not merely axioms: witness closure is what makes the union a model of the entire theory.